Analysis of Arbitrary High Order Spectral Volume Method for Hyperbolic Conservation Laws Over Rectangular Meshes (Journal of Computational Mathematics, 2025)
This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the $L^2$ norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.
Keywords: Spectral Volume methods; Energy stable; Superconvergence; Hyperbolic conservation laws